Resonant transfer networks with reactive loads



A. M. FETTW-EIS 3,501,593

RESONANT TRANSFER NETWORKS WITH REACTIVE LOADS Original Filed Nov. 12,1964 March 17, 1970 3 Sheets-Sheet 1 March 1970 A. 1.. M. FETTWEIS 3, 0,593

RESONANT TRANSFER NETWORKS WITH REACTIVE LOADS Original Filed Nov. 12,1964 3 Sheets-Sheet 2 4-||V4 3 W5 m fie 7 a 4 1A0 a J w I Q U A I i adily United States Patent Int. Cl. H043 US. Cl. 179-15 4 Claims ABSTRACTOF THE DISCLOSURE Resonant transfer networks terminated in reactiveloads are nonetheless made frequency independent by proper selection ofratio of input and load reactances. The transfer characteristic can alsobe tailored to a desired frequency dependency when such a characteristicis desired to counteract load frequency characteristics.

This application is a continuation of 410,533, filed Nov. 12, 1964, nowabandoned.

The invention relates to resonant transfer networks with reactive loadsincluding a air of reactive energy storage devices such as a pair ofcapacitances which may be effectively and repeatedly interconnected by anetwork so designed that with a given energy in the first device,obtained from a signal source which may be coupled thereto, at thebeginning of an effective interconnecting time, a corresponding amountof energy is stored in the second device at the end of said time.

A resonant transfer network of this type is described for example, inthe Proceedings of the Institution of Electrical Engineers (a Britishpublication), September 1958, volume 105, part B, page 449 etc., in anarticle entitled Efficiency and Reciprocity in Pulse-AmplitudeModulation, K. W. Cattermole, and more particularly in paragraph (5.5)of this article. With such an arrangement, if the effectiveinterconnecting time of two capacitances of equal values is equal tohalf the period of natural resonance of the circuit comprising these twocapacitances together With the interconnecting network which is mostsimply realized by way of an inductance, by coupling the firstcapacitance to a signal source, the instantaneous voltages of the lattermay be sampled not on a voltage basis but on an energy basis. Each timethat the resonant circuit is established by interconnecting the twocapacitances with the help of a gate and through a series inductance,the voltage across the first capacitance is transferred to the second atthe end of the interconnecting time and if these times are small whencompared to their repetition period, the changes of voltages across thesecond capacitances may be regarded as practicall instantaneous. Inother words, a so called box-car waveform is produced across the secondcapacitance. If the latter is unloaded but connected for instance to thehigh input impedance of a buffer amplifier, during the time intervalbetween two energy transfers from the first to the second capacitance,the voltage across the latter will remain substantially constant and itmay be analyzed, e.g. coded, in any desired manner. As compared to merevoltage sampling, the above energy sampling into a capacitive loadoffers the advantage that all other things being equal, it is easier torealize a true box-car waveform with fiat portions between the instantsat which the voltage across the second capacitance is abruptly modified.Indeed, the energy in a capacitance is proportional to the capacitanceand to the square of the voltage so that for a given voltage level, thegreater the energy, the greater may be the capacitance and in turn thismeans that for a given relatively high resistance shunting thecapacitance, the greater the time constant will be, thus ensuring a fiatvoltage top across the capacitance.

With the above circuit, the voltage obtained across the secondcapacitance at the end of the transfer time can be precisely equal tothe voltage across the first at the start of this transfer time,provided the two capacitances are of equal value and that there are noresistive elements involved in the resonant transfer circuit. Since thelatter necessarily involves at least a series inductance and a gate inpractice these two elements will bring in some resistive losses.

The invention is based on the insight that such losses not only cause areduction of the voltage across the sampling capacitor, which reductionwill in general be relatively slight and is not particularlyobjectionable, but they introduce a frequency distortion in the overalltransmission. More particularly, the higher the frequency of theoriginal signal the greater will be the loss caused by the resonanttransfer. Such frequency dependent characteristic does not occur whenthe second capacitance of the resonant transfer circuit is connected toa resistive load through a low-pass filter as will generally be the casefor the first capacitance. If when using the energy sampling principleit is desired to avoid a frequency dependent characteristic due tolosses of the resonant transfer circuit, by periodically removing theenergy from the second capacitor with the help of a clamping circuit itis possible to keep a flat response. While this additional expediture isa satisfactory and in fact essential solution in the case of pulse codemodulation systems where the energy sampling process is generallyapplied with a common capacitance connected to a common PCM coder, i.e.used in time division multiplex fashion for a plurality of signalsources, it is not always necessary or desirable to resort to the use ofclamping means, nor in fact possible to do so in some circumstances,i.e. when it is essential to keep the voltage sample across thecapacitance until the next exchange of voltages.

A general object of the invention is to obtain a flat frequency responseusing the resonant transfer principle when the receiving capacitance issubstantially unloaded and this despite the resonant transfer circuitnormally introducing such a frequency dependent characteristic, e.g. dueto resistive losses of the series transfer inductances and of theinterconnecting gate or gates.

Although a fiat response is generally desired, it may happen thatanother part of the transmission circuit incircumstances, i.e. when itis essential to keep the voltage desirable to correct. Also, it maypurposely be desirable to introduce a frequency dependent characteristicas disclosed for example in the US. Patent No. 2,621,- 251 wherepro-emphasis is introduced at the coding end of a PCM transmission inorder to reduce quantization noise, de-emphasis at the decoding endrestoring an overall flat characteristic in function of frequency.

Another object of the invention is to obtain a frequency dependenttransfer characteristic, particularly one where the response increasesas the frequency increases, using the resonant transfer principle withthe receiving capacitance substantially unloaded.

In accordance with a characteristic of the invention, the resonanttransfer networks with reactive loads as initially defined arecharacterized by a ratio between the reactances of said storage deviceswhich differs from unity.

In accordance with another characteristic of the invention, said ratiodifferent from unity is chosen in such a way as to obtain a flatfrequency response in the transfer characteristic between the twostorage devices.

In accordance with another characteristic of the invention, said ratiodifferent from unity is chosen in such a way as to obtain a frequencydependent response particularly one which increases as the frequencyincreases,

3 in the transfer characteristic between the two storage devices.

In accordance with another characteristic of the invention, resonanttransfer networks with reactive loads as specified above arecharacterized in that the negative of said ratio is equal to the inverseLaplace transform at time t where t represents the effectiveinterconnecting time of the two storage devices, of the differencebetween the open circuit impedances at each end of the resonant transfernetwork including the two storage devices, divided by the product of thedifferences between the impedances of said storage devices by p, theimaginary angular frequency.

In accordance with another feature of the invention, resonant transfernetworks with reactive loads as specified above are furthercharacterized in that said storage devices are each constituted bycapacitances interconnected at one terminal, that the capacitanceforming said second device and constituting said reactive load has asmaller value than that constituting the first storage device and thatthe negative of said ratio is equal to the ratio between the differenceof the voltages across said capacitances at the end of said time t andsaid difference at the start of said time t In accordance with yetanother feature of the invention the capacitance constituting saidsecond storage device divided by that constituting the first is equal toc where n1 is the negative real part of the pair of complex conjugateroots of the resonant transfer network, this ratio corresponding to aflat loss substantially independent of frequency and incurred whentransferring a voltage from the first to the second capacitance.

When the sampling capacitor, that is, the capacitor connected to thehigh input impedance buffer amplifier, is of a lower value than thecapacitor that is on the side of the signal source, a flat response isobtained. While the reduction on the magnitude of sampling capacitanceentails an energy loss, nonetheless, it enables a voltage to build up onthis capacitance which at the end of the transferred time is equal tothe voltage originally impressed on the capacitance on the side of thesource at the start of the transferred time. When equivalent losses ofthe resonant transfer circuit are small however as will generally be thecase, the flat loss will be relatively unimportant and all theadvantages of energy sampling are retained while a distortionlesstransmission is secured.

The above and other objects and characteristics of the invention as wellas the invention itself will be better understood from the followingdescription of a detailed embodiment thereof to be read in conjunctionwith the accompanying drawings which represent:

FIG. 1, a general resonant transfer circuit including terminatingresistances and useful to explain the theory on which the invention isbased;

FIG. 2, the actual resonant transfer network N shown as a block in FIG.1 in the particular case of resonant transfer with intermediate storage;

FIG. 3, a so called pulse impedance interconnecting circuit constitutinga transposed equivalent of the actual resonant transfer circuit of FIG.1 and useful to analyze its operation;

FIG. 4, that part of the circuit of FIG. 1 which is effective at highfrequency;

FIG. 5, a circuit equivalent to the general circuit of FIG. 4 using apair of like L-networks;

FIG. 6, a T-network equivalent to the circuit of FIG. 5;

FIG. 7, a 1r-network equivalent to the T-network of FIG. 6;

FIG. 8, a T-network representation of the reactive network N appearingin FIG. 1;

FIG. 9, a T-network representation of the resistive interconnectingnetwork N of FIG. 3;

FIG. 10, a Z-terminal network which may constitute the reson nt tran fernetwork N of FIG. 4 when coupled between terminals 3 and 4 thereof andwith terminals 3' and 4' directly interconnected;

FIG. 11, a resonant transfer circuit without load across the outputreceiving capacitor, and

FIG. 12, a circuit which is a transposed equivalent of that of FIG. 11in the same way as FIG. 3 shows a circuit which is a transposedequivalent of that of FIG. 1.

Referring to FIG. 1, the latter shows a general circuit serving toillustrate the resonant transfer principle which will be analyzedhereafter in order to derive a so called pulse impedance interconnectingcircuit which is represented in FIG. 3, FIG. 9 representing part of thecircuit of FIG. 3 which is shown therein in block diagram form. In turn,this pulse impedance interconnecting circuit will permit to calculatethe transmission performance of circuits such as that of FIG. 1. Whenthat part of the resonant transfer circuit which is effective at highfrequency (FIG. 4), i.e. during the actual transfer time interval,produces transmission losses, this analysis will permit to show that inthe ordinary resonant transfer circuit, when both ends are terminated byresistances (FIG. 1), a flat transfer characteristic can be secured.This is not so however when one of the two capacitive stores of theresonant transfer network is left substantially unloaded by a resistivetermination, this normally leading to a transfer characteristic which isfrequency dependent unless the ratio between the two capacitancesdiffers from unity in a measure which depends on said losses. Moreprecisely, the unloaded capacitance will have a smaller value than thecapacitance on the input side. Although such a choice will permit tosecure a transfer characteristic which is flat in function of frequency,this decrease in the value of the output capacitance will neverthelessmeans an energy loss, but this will be constant irrespective offrequency and provided the losses of the resonant transfer circuit arenot high, this additioinal flat loss will be correspondingly slight.

It will also be shown that a frequency dependent characteristic of theresonant transfer circuit, when using an unloaded capacitance, may besecured on purpose if another part of the transmission circuitintroduces a loss, e.g. an attenuation increasing with the frequency. Inthis case, a suitable ratio between the two capacitances will permit toequalize the transmission characteristic. Pre-e-mphasis may also besecured in the same manner.

In FIG. 1, the blocks N and N are two 4-terminal networks which are notnecessarily the same and which are assumed to contain only constantelements, that is, fixed resistors, inductors and capacitors. On theside of the pair of terminals 3-3 for N and on the side of the pair ofterminals 4-4 for N these two networks N and N are interconnected by wayof series switches, S on the side of N and on the side of N to a networkN also shown as a block and which may in principle contain additionalswitches (not shown in FIG. 1) which like S and S are periodicallyoperated. At its other pair of terminals 1-1, N is fed by a source ofvoltage Ee having an internal resistance R This source is represented inFIG. 1 merely by its complex amplitude E, and the factor echaracterizing the frequency of the signal, p being the complex angularfrequency parameter and t the time, is also omitted for all othervoltages identified in FIG. 1, i.e. V across terminals 1-1, V acrossterminals 33', V, across terminals 4-4 and V across terminals 22' towhich is connected the load resistance R The input impedance of N on theside of terminals 3-3, i.e. next to the switch S is designated by Z, andthe corresponding impedance for the network N across terminals 4-4' isdesignated by Z These impedances Z and Z are assumed to become those ofpure capacitances C and C when the frequency becomes sufficiently high.Accordingly, C and C represented inside the respective networks N and Nby single shunt capacitors across the terminals 3-3 and 4-4respectively, although they may be composed of a plurality of capacitorsincluded in N and N may be identified in terms of Z, and Z which arerespective functions of p by 1 pi ipzam 1 ll mpzitm The network Nforming the resonant transfer network and which in its simplest form maybe constituted by a single series inductance (not shown in FIG. 1) whenthe two energy storage devices are two capacitances such as C and C asshown, will be assumed to be such that the voltages across thecapacitances are sharply modified during the actual resonant transfertime, e.g. during the time of closure of the switch such as Scorresponding to the capacitance C This is obtained by a resonancephenomenon and in the case of the direct resonant transfer with theswitches S and S closedand opened in unison, as well known, the resonanttransfer time t during which the switches are closed may be chosen equalto half the natural period of oscillation of the circuit constituted bythe inductance and the capacitances C C in series. If this transfer timet is sufficiently small with respect to the repetition period T, it canbe justifiably assumed that any other current or voltage in the networksN and N remain practically unchanged during each such briefinterconnecting time.

FIG. 1 also shows the times at which the switches S and S are operated.The recurrence period of the closures is the same for both switches andequal to T but as shown in the timing diagram of FIG. 1, the switch S isclosed at times which lag by T behind the times of closure of the switchS or alternatively which lead such closure times by T so that T=T +TThis is a general timing diagram for the switches S and S and in fact itcorresponds to a resonant transfer circuit using the intermediatestorage principle also described for instance in the article previouslyreferred to and more particularly under paragraph (5.4). In a directresonant transfer circuit, the times of closure of the switches S and Swill coincide so that one of the times such as T will be equal to whileT will be equal to T the repetition period. If intermediate storage isused however, the network N may contain additional reactive storageelements as well as additional switches.

FIG. 2 shows how such a network N may be decomposed when using theintermediate storage principle. As shown within a dotted outline, theresonant transfer network N connected between the switches S and S isnow decomposed into two resonant transfer networks N and N which are onthe one hand connected to the terminals 3-3 through switch S and toterminals 4-4 through switch S respectively, and on the other handinterconnected via additional serial switches S and S Serial switch Sleads to terminal 5 which is directly interconnected with terminal 6 towhich switch S is connected. The networks N and N are further connectedon the inside to terminals 5' and 6 respectively, which terminals arealso directly interconnected. Between the joint terminal 5, 6 and thejoint terminal 5, 6' is an intermediate storage capacitor C which isshunted by a resistance R and which represents a leakage resistance.This permits to take into account variations of potential acrosscapacitor C when both the switches S and S, are open as shown. Further,by means of the additional switch S a further resistance identified by Rmay be coupled across C when switch S is closed. This further resistanceR is not necessarily present in an intermediate storage arrangement butas disclosed in U.S. Patent No. 3,187,100, filed Apr. 24, 1961 it may beconstituted by a negative resistance which will help to keep a constantvoltage across capacitor C during the time intervals when both switchesS and S are open, or even enable an increase of the voltage V across Cin order to secure amplification. The leakage resistance R showndirectly connected across C in FIG. 2 is generally quite high so thatduring the intervals of time for which switch S is closed the combinedparallel resistance across C will thus be practically equal to R only.At other times, when switch S is open, the resistance R' can usually bedisregarded as sufiiciently high.

In the network of FIG. 2, the resonant transfer networks N and N willrespectively permit a direct resonant transfer between the capacitance Cof FIG. 1 and the intermediate storage capacitor C (FIG. 2), and betweenthe latter and the capacitance C of FIG. 1. The first case will happenwhen both switches S and 5;, are closed simultaneously and the secondwill take place when boththe switches S and S are closed simultaneouslyat times which differ from the closure times of the first two switches.

As in FIG. 1, FIG. 2 also shows a timing diagram for such closures andagain, the closure times of the switches 8 lag by T behind the closuretimes of the switches 5 During at least part of the times T and T theshunt switch S may be closed, e.g. to introduce during repetitive fixedtime intervals a resistance R of negative value across capacitor C Itwill be noted of course that such times of closure of the switch S mustnot be deemed infinitely short as the time of closures of the switchessuch as 8 and 8 Further, modification of the voltage V acrossintermediate storage capacitor C may also occur when switch S is closedby a resonant transfer as also disclosed in the US. Patent No.3,187,100.

A general analysis of the circuit of FIG. 1 will now be made, without atfirst specifying a particular mode of operation, i.e. direct transfer(simultaneous closure of S and S or intermediate storage transfer(separate closures of S and S In what follows, it is assumed that bothtransfer times, i.e. times of closure of the switches S and S areinfinitely short. With V and V representing the voltages across Z i.e. Cand Z i.e. C respectively, the voltages V and V may be used to identifythe corresponding voltages just before the closure of the respectiveswitches S and S while V and V may be used to identify the respectivevoltages immediately after the closures of the switches S and S Assumingthat the elements of the circuit of FIG. 1 and particularly those of theactual resonant transfer network N are linear, from a formal viewpoint,the resonant transfer arrangement may be taken mathematically as a meansto realize two linear homogeneous independent relations between themagnitudes V V V and V These two relations may be written as giving thevoltages V and V cxplicity in terms of the voltages before the closureof the respective switches, B B B and B being dimensionless parameterswhich depend solely on the actual resonant transfer arrangement.

FIG. 1 also indicates the complex magnitudes of the currents I I I andL, which flow through the corresponding terminals 1, 2, 3 and 4 eachtime in the direction of the network N and the currents I and 1.; may bedefined by 7 where J and I are respective constants having thedimensions of a current and where the function of time D(t) is definedby D t d t-mT where m is an integer and this function thus correspondsto an ideal train of periodic pulses with a period T, the

function d(t) being the conventional unit impulse having an ideallyshort duration and the inverse dimension of the time t.

Considering the voltage such as V across the impedance Z a relation maybe established between these two quantities and the impedance by which(since V is taken as positive with respect to terminal 3' and since thecurrent I enters Z at the terminal} must be multiplied to obtain V isindependent of J and is a function of t with a period T. As the Fouriercomponents of (7) have all the same complex amplitude l/T, in theabsence of a source B one may write where n is an integer and P is theimaginary angular sampling frequency, i.e.

Evidently, a like relation links V Z and J Since V and V; are functionsof t with a period T, at any instant of closure of a switch such as Swhen considering V the voltages V and V immediately before andimmediately after the considered instant at which the switch S closesare independent of this particular instant, although this would not betrue of the actual instantaneous amplitudes. Considering the sum as wellas the diiference of such voltages as V and V31 the following relationsmay be written down wherein U and U; are new voltage parametersrespectively equal to the half sum of the voltages across Z and Zimmediately before and immediately after the closure of the respectiveswitches S and S while R 1 and R 5 are the corresponding half di erencevoltages across Z and Z respectively, the new parameters R and R beingevidently resistive.

Just as (8) establishes a proportional relation between such voltages asV and such currents as J like relations may be established this timebetween the voltages U and U; as defined by (10) and (11) in terms ofthe respective currents J and J Calling the ratio between U and -J (inthe absence of a source) the pulse impedance Z and with a like pulseimpedance Z linking U and -J.;, the following relations may be writtendown wherein the voltage parameter E, appearing in (14) will bediscussed later. The second-expressions on the right establishing adefinition of the so called pulse impedances Z and Z have been obtainedby considering (8) and the like relation linking V and J as well as aknown theorem by which at a point of discontinuity, a Fourier seriesconverges towards the arithmetic mean of its values just before and justafter the discontinuity. Considering (l4) and (15) the so called pulseimpedances Z Z will be recognized as equivalent to what was introducedin the United States patent application Ser. No. 213,375 filed on July30, 1962 and assigned to the assignee of this application as the averagepulse sequence impedance, itself corresponding to the arithmetic mean oftwo so called pulse sequence impedances previously introduced in theabove mentioned article although in the latter, these quantities had infact the dimensions of impedances divided by the sampling period T.

If the impedance such as Z is the input impedance of a network whichlike N is fed on the other side by a voltage of which the amplitude is Eas indicated in FIG. 1, the voltages just before and just after theclosure of the switch such as S i.e. V and V will no longer be directlyproportional to the current J but they will be linear functions of thiscurrent 1 a constant term E being introduced for the expressions givingV and i This is obtained by a direct application of the superpositionprinciple or what amounts to the same thing, Thevenins theorem in itsgeneralized version. See, for example, p. 74 of Electronic and RadioEngineering by F. Terrnan published by McGraw-Hill, copyright 1955. Thevoltage E appearing in (14) is therefore the open circuit voltagemeasured across terminals 3-3 and due solely to the voltage E (FIG. 1).

The pulse impedances Z and 2 introduced in (14) and (15) have beendefined by these same relations. The resistive parameters R and Rintroduced in (i2) and (13) can be defined in the following manner.Considering for instance the voltage V across the impedance Z at aninstant when S closes for what may be considered an infinitely shorttime, the product of C by the voltage difference V -V is proportional tothe charge brought at that instant by the current I Accordingly, byconsidering (5) the charge C (V V is equai to J T. Therefore theparameters R and R may be expressed this by direct application of (12)and (13).

Due to the relations established so far, it is now possible to introducea so called pulse impedance interconnecting circuit related to theresonant transfer circuit of FIG. 1 and which will facilitate theanalysis of its properties. In this related circuit, instead of thevoltages such as V and V and the currents I and L; which appear in FIG.1, it is now the voltages U and U; as well as the currents J and J whichare used.

FIG. 3 shows this related pulse impedance interconnecting circuit whichuses a pulse impedance 4-terminal interconnecting network labelled Nhaving input terminals 3-3 and output terminals 4-4 by analogy with thecircuit of FIG. 1. But this time it is the voltage U which appearsacross terminals 3-3 and the voltage U which is present across terminals44', while the currents J and J fiow into N through the terminals 3 and4 respectively. The introduction of this 4-terminal interconnectingnetwork is possible due to the fact that from the relations (10), (11),(12) and (13) the voltages V V V and V may be replaced into the twolinear relations (3) and (4), giving wherein the impedance parametersW33 W W and W together constitute the impedance matrix of the 4- as R orwherein B is function of the parameters B B B and B44, Let

Thus for the direct resonant transfer the parameters W and W areresistances proportional to R while the parameters W and W areresistances proportional to RC2.

FIG. 3 shows that the terminals 3-3 of N are fed by a source of voltageamplitude E and of internal impedance Z This is a direct result of (14)which also defines the pulse impedance Z while E was defined as the opencircuit voltage amplitude of the network N of FIG. 1 when solely fed bythe source of voltage amplitude E. Likewise, the pulse impedance Z isshown by FIG. 3 to be connected across terminals 4-4, this beingjustified by 15).

The network of FIG. 3 related to that of FIG. 1 and using the pulseimpedances and the interconnecting network will permit to deriveexpressions for conversion and reflection coefficients which will definethe operation of the overall circuit of FIG. 1. In the latter, it willbe recalled that all the voltages V V V and V are complex amplitudeswhich depend on the sampling frequency, the multiplying factor e havingbeen omitted throughout, this factor affecting the input source shown inFIG. 1 of which only the amplitude E has been indicated. Thus,considering V which will be of particular interest in assessing aconversion coeflicient for the transmission between terminals 1-1' and2-2, this can be written as a function of the time t as V (t) V e 2 ri s2 25 wherein P is the imaginary angular sampling frequency previouslydefined by (9). The current I (t) can be defined in exactly the same wayas V (t), or in other words, one may write linking the component oforder n contained in V with the component of like order contained in I Aconversion coefficient of order n may then be defined by analogy withthe classical theory for constant parameter networks. In the latter, thesquare of the modulus of the conversion coefiicient may be defined asthe ratio between the power in the load resistance, i.e. R and themaximum power which can be obtained from the source E. Since the firstis equal to the square of the modulus of the voltage component V oforder it across R divided by this resistance, while the second is equalto the square of the modulus of E divided by 4R a conversion coefficientof order 11 characterizing the transmission from terminals 1-1' to 2-2may be defined as where the second expression is immediately obtained bya direct application of (26) The voltage amplitude, V (t) expressed alsoas a function of the time It may evidently be written in the same way asV in (25), i.e.

A reflection coefiicient of order n, i.e. S may then be defined by ln I1n l S11v2 E 2 E 29 where the second expression in terms of I is readilyobtained by considering the voltage across terminals 1-1' in FIG. 1, Ibeing evidently the complex current amplitude of order 11 correspondingto the complex voltage amplitude of order n Thus with this definition ofthe reflection coefficient of order n, the latter will be Zero wheneither the corresponding complex voltage amplitude or complex currentamplitude is zero. The definition given by (29) is however valid onlywhen n is distinct from 0. In the latter case, the reflectioncoefiicient of order 0, i.e. S may be written as clearly showing thatthis particular reflection coefficient will be zero when the complexamplitude voltage V is equal to E/2.

The complex current amplitudes of order n such as 1 and I which appearin the second expressions given for (27) and (29) as well as the complexcurrents amplitude I which appears in the second expression given by(30) may now be calculated in terms of the equivalent circuit shown inFIG. 3. For the current amplitude I it should be noted that thisconsists in the linear superposition of the current due to I and thatwhich would be due to the source of amplitude E if the terminals 3-3were continually open circuited. For the complex current amplitude Ihowever this is solely depending on J i.e. Equation (5). Thus, I may beexpressed as E i 1o=m+Ja 1(P) (31) where the first term gives thecurrent due to the source I, Z representing the open-circuit impedanceof network N measured across terminals 1-1'. The second term is equal tol multiplied by M (p) which is the current transfer coefficient of the4-terminal network N from terminals 3-3 to terminals 1-1'. When thenetworks such as N and N of FIG. 1 are reciprocal, such current transfercoefficients as M (p) for N are equal to the opencircuit voltagetransfer coefiicients in the opposite direction, i.e. from terminals 1-1to terminals 3-3 for N This well known relation will be quite clear whenconsidering FIG. 8 which represents the network N of FIG. 1 as anequivalent reciprocal T-network fed by the source E with its resistanceR; across terminals 1-1 and left open-circuited at terminals 3-3. Withthe series impedance branches connected to terminals 1 and 3respectively labelled Z11-Z13 and Z -Z and with the shunt branchconnected to the directly coupled terminals 1' and 3 labelled Z it isreadily shown that the opencircuit voltage E across terminals 3-3 andalready referred to in relation to (14) may be expressed in temn of E bythe current transfer coefficient of N from terminals 3-3 to terminals1-1. For all other complex amplitudes of of equations (17) and (18) aswell as (14) and These currents may thus be written and the conversioncoefi'icient identified by (27) as well as the reflection coefiicientsidentified by (29) and (30) can now be expressed in terms of the pulseimpedances Z and 2 the W parameters and the coefficients M (p) and M(p+nP), this with the help of such equations as (34) and (36) whenapplied to find a modified expression for S given by (27). For thisconversion coefficient of order n from terminals 1-1 to terminals 2-2,equation (27) thus becomes The interconnecting network N of FIG. 3 willnow be more specifically identified in terms of the actual resonanttransfer network N and the capacitances C and C of FIG. 1 by consideringthe particular case of the direct resonant transfer. For the directresonant transfer system, the network N of FIG. 1 is without memory sothat there is no stored energy in this network at the beginning of atransfer period. Moreover, the case of the direct resonant transfermeans that the two switches S and S operate simultaneously and remainclosed during precisely the same time interval, i.e. 2

The absence of energy in the network N at the beginning of each transferperiod when the switches S and S are simultaneously closed, i.e. with Tof FIG. 1 equal to zero may be secured in various ways. A firstpossibility is to realize the network N in such a way that all itselements, e.g. the highway capacitance, are exactly discharged at theend of the transfer period when the two switches S and S are reopened,so that they will certainly also be discharged at the beginning of thenext transfer interval. A second method consists in realizing thenetwork so as to cause a practically instantaneous discharge from theopening of the switches, or in any event a practically completedischarge between the end of a transfer period and the beginning of thenext. Finally, a third method consist in foreseeing inside N auxiliaryswitches which cause the desired discharge during a period of timesuitably chosen between two successive transfer periods. In this casethere is nearly always an advantage in operating these auxiliaryswitches in a periodic manner just as S and 8;. This is however notstrictly necessary since the overall result is the same as soon as thedischarge is complete at the beginning of the next transfer period.

The first of these three methods is the more interesting since it is notaccompanied by a loss of energy. It is not however possible to realizeit perfectly due to the inevitable tolerances on the values of theelements as well as on the timing for the closure of the switches S andS For this reason, in practice a combination of the first with at leastone of the other two methods will be used.

For such a direct resonant transfer arrangement it is now possible tocalculate general expressions for the dimensionless B parametersappearing in the relations (3) and (4).

Referring to FIG. 4, the latter shows the 4-terminal network N of FIG. 1when the switches S and S which are not shown in FIG. 4 are closed sothat the two capacitances C and C respectively across terminals 33' andterminals 4-4 are now also directly in shunt across the two ends of N Itis clear that at high frequency, the network N cannot be capacitive atits input and output across terminals 3-3 and 4-4', since otherwise acontrolled lossless resonant transfer from capacitance C to capacitanceC and vice-versa would be excluded. Thus, at high frequency for theresonant transfer, the 4-terminal network of FIG. 4 is constituted atthe terminals 3-3 solely by the capacitance C and likewise at theterminals 4-4' solely by the capacitance C Hence, the analysis of thecomplete circuit of FIG. 4 may be performed by assuming that at theinstant the switches are closed, ideal impulses of infinitely shortduration, of infinitely large amplitude and of moment v C and 11. C areapplied across terminals 3-3 and 4-4' respectively. Indeed, if V and vrepresent the instantaneous voltages before the closure of the switchesS and S and thus corresponding to the voltage amplitudes V and V so farconsidered, these ideal impulses will instantly carry the instantaneousvoltages across terminals 3-3 and 4-4' to the respective requiredvalues. The instantaneous voltages v and v corresponding to the voltagesV and V so far considered are related to the instantaneous voltages V31,and v by two linear equations which have coefficients depending on thecharacteristics of the complete network of FIG. 4. More precisely, sinceit has been assumed that the network N was without energy at the instantthe switches were closed the instantaneous voltages v and v can becomputed in terms of the in verse Laplace transforms at time t i.e. thetransfer time during which the switches are closed, of the impedancematrix elements of the complete network of FIG. 4. The coeflicients 'bywhich the instantaneous voltages v and V4}, must be multiplied toproduce the instantaneous voltages V and v by linear combinations arethe parameters B B B and B of equations (3) and (4) since all theinstantaneous voltages such as v are related to the correspondingvoltage amplitude V by the same proportionality factor. Thus, for thedirect resonant transfer system when N is without energy at the time ofeach transfer interval and calling L the inverse Laplace transform attime t the parameters B B34, B and B are given by The 4-terminalcomplete network of FIG. 4 is reciprocal which means that out of thefour impedances defining its impedance matrix and appearing in the abovefour equations, i.e. Z Z Z and Z the last two are equal to another. Inview of (39) and (40) this means that B /B is equal to C /C and in turn,in view of (16) and (17), equal also to R /R Accordingly, for the directresonant transfer, by considering (20) and (21), when the 4-terminalcomplete network of FIG. 4 is reciprocal, i.e. Z =Z the equivalentresonant transfer network N of FIG. 3 is also reciprocal, i.e. W W Sincethe parameters B are now defined by the equations (38), (39), (40) and(41) the resistive parameters W expressed by (19), (20), (21) and (22)can be calculated. These are pure resistances directly proportionaleither to R (for W and W or to R (for W and W The relations between theresistive parameters of the equivalent resonant transfer network N, ofFIG. 3 and those of the complete overall network of FIG. 4 comprisingthe network N associated with the shunt capacitances C and C can befacilitated by considering alternative representations for the networkof FIG. 4.

FIG. 5, shows a circuit equivalent to that of FIG 4 and which comprisesthe association obtained by inserting between terminals 33' and 4-4 acascade arrangement of two like 4-terrninal networks N and N the secondbeing however reversed in direction with regard to the first and beinginserted between two ideal transformers TR and TR having a voltage ratioequal to C /C as will be justified later. While these two idealtransformers are identical, they are coupled in opposite ways, i.e. asstep-up and step-down transformers respectively, so that any impedancepresent at either the terminals 3-3' or 4-4 is seen unchanged across theother pair of terminals 44' or 33. The two networks N and N areidentical L-networks each comprising a series impedance Z followed by ashunt impedance Z -Z so that Z and Z represent respectively theopen-circuit and short-circuit impedances of the network such as Nmeasured on the side of the series impedance Z away from the shuntimpedance Z Z The 3-parameter, i.e. Z Z and C C reciprocal network ofFIG. 5, can be readily transformed into the equivalent T-network of FIG.6 by eliminating the ideal transformers TR and TR of FIG. 5. Theimpedance parameters Z Z34, Z and Z of the networks of FIGS. 4, and 6are therefore identified by Bearing in mind that at high frequency, notonly must the open circuit impedance Z measured at terminals 3-3 beequal to that of the capacitance C but also the short circuit impedanceZ and that such impedances measured at high frequency across terminals44 must necessarily be respectively, it is clear that the choice of forthe voltage ratios of the ideal transformers TR and TR as shown in FIG.5 is justified.

The network of FIG. 6 thus constitutes a general representation of areciprocal 4-terminal network, with the sole restriction that the twoseries arms are impedances of like nature since at high frequency, whenboth Z and Z are reduced to the impedance of C the ratio between theimpedances of these two series arms, i.e. Z Z and Z Z =Z Z should bethat between C and C FIG. 7 represents a vr-network equivalent to the T-network of FIG. 6 and may be readily obtained from the latter byconventional transformations or by starting from a circuit analogous tothat of FIG. 5 but wherein the networks N and N are reversed, e.g. for Nthe shunt impedance Z Z is now directly in shunt across terminals 3-3.The network of FIG. 7 facilitates the identification of Z and Z with theactual elements of the complete resonant transfer network of FIG. 4 andfor the simpler resonant transfer circuit of the direct type it isreadily seen that Z identifies itself with 1/pC the impedance of thecapacitance C 14 Making use of the inverse Laplace transforms at time tof the impedances Z and Z so defined and of (42), (43), (44), theparameters identified by (38) to (41) can now be expressed at which notonly define the parameters B and B as inverse Laplace transforms of Z,and Z respectively each multiplied by C but also give a relation betweenthe four parameters B B B and B due to the networks of FIGS. 5, 6 and 7being reciprocal, i.e. Z =Z This condition between these four parameterspermits to write a new expression for the parameter B given by (23), infunction of B and B i.e.

Additionally, (50) expresses B as a function of Z Z this by using (42),(43) and (44).

Returning to the conversion coefficient of order n, i.e. 8 defined by(37) and specifying the transmission from terminals 1-1 to terminals 2-2for any component of the various sidebands which are obtainable at theoutput of N (FIG. 1) depending on the passband of the latter, theparameters W W34, W and W appearing in (37) and defined by (19) to (22)can be expressed in terms of the B and B dimensionless coefficients. Onthe other hand the open circuit voltage transfer coefficients M and Malso appearing in (37 can be expressed in terms of the resistive part ofthe impedances Z and Z respectively and. in terms of the respectiveterminating resistances R and R This last can best be explained inrelation to FIG 8.

Considering FIG. 8 already described in relation to (32) defining theopen circuit voltage E, at terminals 3-3 and assuming that p is a pureimaginary angular frequency so that w=jp is real, the square of thecoeflicient M may in view of this last equation be expressed as giving Mas a product of three factors, the last of which can be recognized asthe reflection coefiicient at terminals 1-1', i.e. h given by If thenetwork N is purely reactive as is usually the case, Z Z and Z are allpurely reactive and accordingly the second factor of (52) represents thecomplex conjugate M of M Thus (52) may be written as If the impedancesof N are purely reactive, the input impedance Z, measured acrossterminals 33"in the direction of N may beiwritten as in which the secondexpression for Z is readily obtained by replacing eaeh impedance such asZ in function of the corresponding reactance such as 'X and wherein thethird expression for Z iidentifies' its resistive part as R and itsreactive part as X By transforming the second expression for Z the valueof R is found to be R R =MH-R 7 3 RIZ+XII2 1 l i 1 In association with(54) the latter thus gives I and since by virtue of Z being purelyreactive, i.e. equal to jX the modulus of I1 is equal to unity, thismeans that the square of the modulus of M can be expressed directly asthe ratici'of R and R i.e.

L 2 R1 A like expression can be secured for the square of the modulus ofM which is thus equal to the ratio between R the resistive part of Z4,and R the terminating resistance across terminals 2-2. But it should beremembered that whereas RQis a function of w, R willbe a function of asis clear from (37) W In order to transform this expression giving 8 itis still necessary to have the parameters W W34 W and W expressed interms of the dimensionless coeflicients B and B as well as theresistances R and R previously defined.

FIG. 9 shows the equivalent resonant transfer network N introduced inFIG. 3, in the forrn of a T-network of which the series resistances areW W on the side of terminal 3 and W W on the side of terminal 4. Theshunt resistance is W W by virtue of the equivalent network N beingreciprocal as previously explained. Making use of the Equations 45 to 48and 51 into 19 to 22, these three resistances shown in FIG. 9 can beexpressed as W It, is now possible to obtain a very simple expressionfor the conversion coefficient of order n, i.e. 8 given by 37) andparticularly in the case of the direct resonant transfer for which theexponential term contained in this last expression disappears since T(FIG. 1) is equal to 0 as the two switches S and S are simultaneouslyoperated. a r 7 Considering the equivalent circuit of FIG. 3 with thestructure of N given by the T-network of FIG. 9, the ideal situation isto have last network reduced to mere interconnections between terminals3 and 4 on the one hand and terminals 3' and 4' on the other, In thiscase the open circuit voltage source E of FIG. 3 is merely zero. Thiswill be shown below.

feeding the two pulse impedances Z and Z in series. These two pulseimpedances can be decomposed into a resistive and a reactive part suchas R and iX for Z and the resistive parLR of the pulse" impedance Z canbe expressed in function of the resistive part R of Z in the same wayas'Equation 14 gives the pulse impedance Z as a series expressed interms of Z i.e.

As expressed by the above, R must be at least 'equal to R (w)representing the resistive part of the impedance Z A like reasoning canbe made for the resistive part R of the pulse impedance Z which must atleast be equal to '7 25m 1.1/4 (co-i- T W representing the resistivepart of the impedance Z As stated before, in the ideal conditions, thenetwork of FIG. 9 should disappear to permit a direct interconnectionbetween the pulse impedances Z133 and Z in FIG. 3 and these conditionswill be attained' when W5 W =W and W are equal to one another and infact infinite since the two series branches of the T- network of FIG. 9can be replaced by a short-circuit when B =-1 while the shunt branch canbe replaced by an open circuit when B is equal to 1. Considering (*37),in such a case, by dividing both the numerator and the denominator by 'Wonly Z +Z will be left in the denominator. For a maximum value of S 5this denominator should be minimlim and this will be the case if boththe resistive part of Z +Z and the reactive part of this combined pulseimpedance are minimized. This will be true when are satisfied, 63) and(64) in view of the minimum value for the resistive part such as R gbeing equal to R (w) as shown (62) If the above three relations aresatisfied in addition to (58) and a like relation for the modulus of Mby replacing into (37) it is found that the modulus of $2111 may reach amaximum value of unity when R is equal to R Filters such as the networksN and N of FIG. 1 which satisfy such conditions may therefore be termedideal filters and ;more specifically ideal filters for single sidebandmodulation by direct resonant transfer since the direct resonanttransfer is the particular case which has been considered immediatelyabove while it is also a single sideband modulation which has beenenvisaged. Otherwise, for double sideband modulation one would have toconsider two conversion coefficients S and S21?n of order n and nrespectively whose moduli should be eqaal to another. 7

Consideration will now be given to the value of the conversioncoefficient 8 in: the case of intermediate storage resonant transfer.For important practical applications of the resonant transfer principlein telecommunicationrgexchanges, it may be desirable 'fthat somecommunications established by any given station should be made inaccordance with the direct resonant transfer principle while othersshould be made following the intermediate storage principle. Whenconsidering also intermediate storage resonant transfer, instead ofideal filters satisfying (63) (64) and (65), so caiied universal idealfilters can then be defined. These have the further properties that bothR and R are equal to the same constant resistance while both X g and Xare equal to Considering FIG. 2 which has already been referred to andwhich describes the resonant transfer network N in more detail in thecase of an intermediate storage transmission, the amplitudes I and I ofthe currents entering the networks N and N through the switches S and Srespectively are still given by (5) and (6). Likewise, the amplitudes Iand I of the currents flowing respectively through terminals 5 and 6towards the centre of the network and formed in like manner by modulatedpulse trains can be defined by While V designates the voltage amplitudeacross the intermediate storage capacitance C the value of V just beforethe arrival of a pulse of 1 due to the closure of switch S can bedesignated by V and in like manner V can be used to identify the valueof V immediately after such a pulse. In a similar fashion, V and V canidentify the values of this same voltage amplitude V just before andjust after the arrival of a pulse of I through terminal 6, i.e. uponswitch 8.; being closed. Keeping the same previous definitions for thevoltage amplitudes V V V and V it is clear that the previous analysisestablishing relations between these four voltage amplitudes on eachside of network N e.g. (3) and (4) and derivations thereof, remain validin the case of the corresponding voltages on each side of the network Nas well as for the corresponding voltages on each side of the network NIn the case of the network N the relations will be between the voltageamplitudes V V V and V whereas for the network N the relations will bebetween the voltages V411, V V4]; and vg Additional relations of thetype given by (10), (ll), (12) and (13) may also be written, this timein connection with the voltage amplitudes V V V and V These are theselast two by taking into account currents J and J (not shown) flowing indirections corresponding to those of 1 and I respectively and byremarking that V -V and V -V can only depend on I and I respectively.The first two above relations introduce the auxiliary voltage parametersU and U and the last two introduce also the resistance R which in thesame manner as R and R i.e. Equations 16 and 17 is given in function ofthe capacitance C by Remembering that the switches S and S close inunison and that is also true for the switches S and 8., but with a timelag equal to T or in other words that the closure of the switches S andS leads the closure of the switches S and S by a time equal to T since T+T is equal to the sampling period T, expressions for the voltageamplitudes V and V immediately before the closure of the respectiveswitches S and 8,; can be derived. Such voltage amplitudes immediatelybefore the closure of the switches, i.e. V and V will be directlyproportional to the respective voltage amplitudes immediately after theclosure of the opposite switch, i.e. V and V respectively, i.e.

Since FIG. 2 shows resistances such as R' permanently connected acrosscapacitance C and corresponding to the leakage resistance as well asanother resistance R which may be temporarily connected across such acapacitance when the switch S is closed, the above equations showexponential terms which include not only a term pT or pT respectivelycorresponding to the time delay in the operation of the switches S and Sbut in addition a term a or a corresponding to the attenuation producedby such resistances during the time interval separating the closure ofthe switch S from the closure of the switch S and vice versa. Clearly,during the time that S is closed the time constant for the intermediatestorage capacitance C will be equal to the product of that capacitanceby the parallel combination of the resistances R and R' while during therest of the time interval separating the closure of a switch S from theclosure of a switch S while switch S remains open, it will be equal tothe product of C solely by the resistance R' Accordingly, such constantsas a and a appearing (73) and (74) are readily calculated by dividingthe corresponding time intervals by the time constants.

Two linear relations corresponding to (18) and (19) may now beestablished this time between the voltages U and U and the currents Jand J with the help of (68), (69), (70), (71), (73) and (74). These arewherein the impedance parameters W W56: W and W are identified byW55=Wss=Rc0c0th FEW-gig (77) Accordingly, the arrangement connectedbetween the terminals 55' and 6-6 of FIG. 2 corresponds a transposedequivalent quadripole in the same way as the equivalent network N ofFIG. 3 corresponds to the actual resonant transfer network N of FIG. 1,and as indicated by (77), this transposed equivalent 4-terminal network(not shown) is symmetrical since its impedance parameters W and W areequal to one another. On the other hand, such a transposed equivalentnetwork or pulse impedance interconnecting network as previously definedin relation to FIG. 3 is not generally reciprocal since this would implythat W should be equal to W and (79) shows that this is not necessarilythe case.

In order to find the conversion coeflicient 8 in the case ofintermediate storage resonant transfer, it is however such impedanceparameters as W 3, W34, W and W identifying the overall transmissionbetween terminals 3-3' and terminals 4-4, i.e. (18) and (19) which mustbe found. These can be calculated in terms of like parameterscorresponding the networks N and N of FIG. 2 as well as in terms of theparameters W W W and W which relate to the central part of the network NHowever, for the present purpose of finding the conditions which must besatisfied by so-called universal filters which can operate equally wellfor resonant transfer as well as for intermediate storage resonanttransfer transmissions, it may be assumed that the direct resonanttransfer networks N and N of FIG. 2 do not introduce any losses so thatU is equal to U while U is equal to U Also, in view of the directions ofthe currents indictated in FIG. 2, the related currents J 3 and J 5 areequal to one another and this is also true of the related currents J andJ This means that (75) and (76) directly lead to By direct analogy with(37), the conversion coefficient defining the transmission fromterminals 1-1' to terminals 2-2 may now be written in terms of theimpedance parameters defined by (77), (78) and (79), i.e.

If the source shown in FIG. 1 to be connected across terminals 1-1' isnow assumed to be connected across terminals 2-2, the conversioncoefficient S characterizing the transmission in the reverse directionmay be written as In the above the denominator does not differ from thatof S given by (82) in view of its symmetry and in the numerator, Wreplaces W M is a function of p -l-nP, M is a function of p, in view ofS characterizing the reverse sense of transmission. Also, recalling thatthe factor eappears in the expression (34) for 1 but not in theexpression (33) for I this factor disappears in the numerator of theexpression (83). Since it will be the modulus of and that of S whichwill actually be of interest to characterize the transmissions, in viewof such an equation as (58) expressing the square of the modulus of M asthe ratio between the resistance R and the resistance R and bearing inmind the remark after Equations 63, 64 and 65 to the effect that Rshould be equal to R in the case of an ideal filter for single sidebandmodulation by direct resonant transfer, it is clear that the differencebetween S and S depends on the value of the factor lZn Wi enPT Thisfactor may be written as Tz-T 1)n D+nP) 2 in which the first expressionfor the factor follows from (79) after some re-arrangement of the termsappearing in the exponent and bearing in mind, e.g. FIG. 2, that T isequal to T +T The second expression follows by recalling that (9)identifies PT/2 as in so that i is equal to +1 or -1 depending onwhether n is even or odd. Also, in the second expression (84) a has beenmade equal to a This last condition can readily be satisfied if R' islarge enough. Then, the dimensionless parameters a and a are either bothzero if R and S are absent or they can readily be made equal to oneanother if S is closed during appropriate lengths of time occupying partof the intervals T and T respectively, i.e., equal lengths of time ofthe same resistance R is introduced by the closure of S both during Tand T The resistances R introduced across C during part of the timeintervals T and T may either be positive in which case a and a are alsopositive, or they may be negative in which case a and a are negativewhereby the voltage amplitude V may be increased instead of decreasedduring such time intervals separating the closures of the switches S andS Considering the second expression of (84), it is thus clear thatprovided a is equal to a including the particular case when both areequal to 0, the only difference between S and S12n is a delay for thesignal of frequency corresponding to p and a like delay for the signalof frequency corresponding to p-l-nP, such delays being independent offrequency.

Thus, while an intermediate storage resonant transfer circuit like thatof FIG. 2 is not a reciprocal circuit arrangement since this would implystrict equality between the conversion coefficients S and 812 the onlydifference between these two coefficients characterizing transmissionsin reverse directions is only a delay and in general, such circuitswhether they involve variable elements as for the intermediate storagecircuit of FIG. 2, or not, will be termed quasi-reciprocal. Assumingthat the intermediate storage resonant transfer circuit of FIG. 2 isquasi-reciprocal so that a is equal to a and more particularly sinceideal conditions have to be determined that both a and a are equal tozero, the impedance parameters W W and W identified by (77), (78) and(79) may be replaced into (82) giving The first expression which isgiven readily follows from (85) using (58) for the equare of the modulusof M and a like relation for the square of the modulus of M while p isreplaced by jw. For abitrary values of the real and imaginary parts ofthe pulse impedances Z and Z of R and of w, i.e. for any given value ofthe denominator of the first expression in (86) the latter will bemaximum if (63) and (64) are satisfied. Then the first expression may betransformed into the second in which the denominator is now written outas a first term equal to the numerator plus a second term which is aperfect square. Clearly then, the modulus of 8 will again be maximum andin fact equal to unity when this second term in the denominator is equalto 0. Equating real and imaginary parts of this second denominator term,one obtains C0( p3 14) cos T RCARDFRM) p4 pa pa p4) n (87) C p3+ p4) (RR 4+X X 4R tan 2 But when previously considering ideal filters forsingle sideband modulation by direct resonant transfer it has alreadybeen shown that R should be equal to R while thesum of X and X should beequal to zero, i.e. (65). In such a case, the above two relationsdirectly lead to Thus, filters such as the networks N and N of FIG. 1Which possess th property that the square of the modulus of their opencircuit voltage transfer coefficients are defined by relations such as(58) while their pulse impedances such as Z are purely resistive andequal to a constant resistance in the passband may be termed universalideal filters or more specifically universal ideal filters for singlesideband modulation with resonant transfer. They will be equallyeffective whether the direct resonant transfer principle is used orwhether intermediate storage is applied as described immediately above.

An analytical expression for a pulse impedance, such as Z may beobtained by considering that an impedance such as Z.,,( p) may bedefined analytically by N Bi AP) T 1:1 (p p where N represents thedegree of Z (p) and B, are resistive constants so that using (17), (2)may be written clearly showing that with Z (p) a function of pT/Z, thepulse impedance is a function of tanh pT/Z, the third form of Zemphasizing the fact that it is an analytical function of tanh pT/ 2.

When the transformed variable tanh pT/2 is equal to unity, (93) and (92)indicate that Z is equal to R Using then a known theorem on boundedfunctions and transposing it to positive real functions, it can beproved that if a pulse impedance such as Z is equal to a constantresistance for a given frequency interval having a length distinct fromzero, then this constant value must necessarily be that particularconstant value R to which the pulse impedance Z d, is equal for aparticular value of the variable tanh tanh 1 If the networks N and N ofFIG. 1 are such universal ideal filters for single sideband modulationby direct 22 together with (58) and a corresponding relation for M aresatisfied within the respective passbands of the filters, thenremembering that T is equal to zero for the direct resonant transfer,the modulus of 8 defined by (37) can be expressed in the general cas ofthe equivalent network of FIG. 9 in terms of the parameters B and BS as2 (96) where the very simple second expression simply equal to half thedifference between B and B is obtained by making use of (94), (95)together with (59), (60), (61) and finally (S8) for the modulus of M aswell as an analogous expression for the modulus of M The dimensionlessparameters B and B defined by (49) and (50) are similar to suchparameters as B previously defined by (38) and the modulus of B as wellas that of B cannot exceed unity. This is true if the 4-termina1 networkN of FIG. 5 is passive. One must note first of all that the impedances Zand Z characterizing this 4-terminal network and on which B and Brespectively depend, each become the impedance of the capacitance C athigh frequency. Thus, considering for instance the impedance Z andassuming that it is devoid of energy, if at a given instant a currentimpulse is applied thereto such that the voltage at its input terminalsi.e. across C is instantaneously carried to unit value then at a time tlater the voltage across the same terminals will be precisely given by Bi.e. (49). Thus, the fact that Z is passive leads to the modulus of Bbeing smaller or equal to unity and an identical relation is true for Bin view of Z being also passive.

If the resonant transfer network of FIG. 1 is unbalanced, i.e. withterminals 3' and 4' directly interconnected, the instantaneous voltage vbetween terminals 3 and 4 immediately after closure of the switches Sand S can be expressed in terms of the instantaneous voltage v betweenthese terminals immediately before opening of the switches by a= 3a 4a33 43) 3b+( 34 44) 4b s( 3b 4b) S b The first expression follows fromthe definition of v,,, v and v being the instantaneous voltages acrossterminals 33' and 4-4 respectively afterclosure of the switches. Thesecond expression is obtained with the help of (3) and (4), v and 11being the instantaneous voltages across terminals 3-3' and 4-4'respectively before closure of the switches. The third expression isderived by using (50) and the last follows from the definition Of VLikewise, with q,, and q the respective instantaneous total chargesacross C and C after and before closure of the switches:

by again using (3) and (4) as well as (45) to (48).

Thus, since neither the modulus of B nor that of B can exceed unity forpassive networks, as expected, the moduli of v and q cannot exceed thosefor v and q respectively.

When the resonant transfer network of FIG. 4 does not lead to suchvalues as 1 and l for the dimensionless parameters B and B respectively,these parameters are neverthless constants, so that the loss (forpassive networks) is constant and independent of the frequency.

A general theory of resonant transfer transmissions has now beenestablished. This covers not merely reciprocal circuits such as directresonant transfer circuits but also so-called quasi-reciprocal ciruitssuch as intermedi- 23 ate storage resonant transfer circuits where thetransmissions in the two opposite directions differ only by apre-determined delay.

FIG. 10 shows a Z-terminal network illustrating the above and consistingof an inductance L shunted by a resistance R with this combination inseries with a further resistance R and as indicated, this Z-terminalnetwork is connected between terminals 3 and 4. Thus, it may besubstituted for the general network N of FIG. 4 and the complete networkof that last figure is thus composed solely of the capacitance C and Cfor the twoshunt branches between terminals 3-3 and 4-4 re-i spectivelyand of the' Z-terminaI network of FIG. 10 for the series branch betweenterminals 3 and 4, terminals 3 and 4' being directly coupled to oneanother. Relating such a practical resonant transfer network to thegeneral equivalent circuit of FIG. 7 which uses the pararn eters Z and Zin addition to C and C one may write The first of these directly givesZ5, while the second may be used to derive an explicit value for Z using(9 9) and more precisely an explicit expression for C 2 1.e.

(P-bmV-bm wherein auxiliary parameters n m and (1 have been introducedto replace the original parameters R', R and L as well as the combinedcapacitance of C and C in series. These parameters, n al and a; areidentified by,

Thus, since Z is constituted by a mere capacitance, i.e. C the inverseLaplace transform of C 2 is equal to unity but for B to obtain the idealvalue 1 the parameter 11 depending on the losses will be made equal tozero. This implies that R (FIG. 10) is equal to zero whilesimultaneously R is infinite. In such a case w t is an odd multiple of1r preferably equal to 11'. If only this last condition for the angularfrequency (0 is satisfied, then B will be equal to e 1 1 and accordinglythe larger the losses of the circuit jof FIGJ'IO, the more B will divertfrom the ideal value of -1 towards 0. H

Thus, while losses in the resonant transfer network N of FIG. 1 willreflect themselves in an overall transmission loss for this circuit usedfor direct resonant transfer with simultaneous closures of the switchesS and S when the two capacitances are connected to filters N and Nloaded by resistive terminations R and R respectively, the overalltransmission characteristic will not be made frequency dependent: It canalso be shown that this re- 24 a mains true for an intermediate storageresonant transfer circuit (FIG. 2). l, 7

But it will now be demonstrated that this is not the case for a circuitusing the capacitance C unloaded, e.g. for the purpose of energysampling.

FIG. 11 shows such a circuit which is of the direct resonant transfertype mentioned at the beginning of the description and this circuit isanalogous to that already represented in FIG. 1 except that the resonanttransfer network N and the switches S and S are here materializetf bytheelectroni'c gate S which replaces the two switches S and S and whichis sirrfply in series with the inductance L. However, there is no filternetwork N and no resistive termination R so that the terminals 4-4 areunited with the terminals 2-2'. Whatever circuit (not shown) isconnected across capacitance C will be assumed to have a high inputimpedance such as that offered by a buffer amplifier. For such a circuitas that of FIG. 11, as previously considered, the impedance ofcapacitance C is the impedance Z found in FIG. 7 while the seriesimpedance between terminals 3 and 4 otfthat figure corresponds to thatof the series inductance L in FIG, 11. Hence, (105) is satisfied whichmeans that for the particular network of FIG. 11, the interconnectingcircuit of FIG. 9 is reduced to a simple series resistance betweenterminals 3 and 4. W

FIG. 12 shows the pulse impedance interconnecting circuit correspiondingto that of FIG. Hand on the basis of the representation given in FIG.31' for the general case. Hence, the equivalent open circuit voltage Eof network N is simply in series with the pulse impedance Z followed bythe equivalent series resistance between' terminals 3 land 4 and finallyby the pulse impedance Z which is simply the pulse impedance of thecapacitance C In the absence of a load across capacitance C thepreviously defined conversion coefi'icient S losses its significance, Avoltage conversion cbefiicient 8 can however be defined as the ratiobetween the voltage V and the voltage E, V having previously beendefined in relation to (4). This voltage conversion coefficient may bewritten 1 V e-ipi-ifgz (RC1+RC2) (107) In the above, the firstexpression is that of the definition, the second follows immediatelyfrom the difference be: tween the expressions (l1) and (13), the thirdexpres; sion for S is due to' the sum of J andlL; being obviously zero,the foi rth is obtained from (15) while the fifth arid last expressionfor 8,, follows from the replacement of 1;, by its value in terms of theelements of the equivalent circuit of FIG. 12. In this last expression,the ratio between the open circuit voltage E and the source voltage Emay be eliminated in terms of the open circuit voltage transfercoefficient M from terminals 1-1' to terminals 3-3, ie (32) and (57) sothat if'an expression for the square of the modulus of S, is sought,this will be'proportional to R /R This leaves the two pulse impedances 3and Z as the two quantities still to be considered in (107).

If the network N is realized as a universal ideal filter of the typepreviously discussed so thatwhile the open circuit voltage transfercoefiicient M is equal to the square root of the real part R of the opencircuit input impedance Z divided by the terminating resistance R thepulse impedance "2 is, in the passband, equal to this pure resistance Rwhich as previously mentioned, must necessarily be equal to theresistance R 25 The pulse impedance Z corresponding to the impedance ofthe capacitance C is readily obtained from (93), N being equal to 1, pto and B to R in view of (92), i.e.

1 tanh Using these results, the following expression for the square ofthe modulus of this voltage conversion coefficient S can be obtained afiat transmission characteristic can be retained as in the case of nolosses when B is equal to 1. While this means that the output voltageacross terminals 4-4 can be kept equal to the value corresponding to thecase of no losses, there will nevertheless be an energy loss due to theresistive elements of FIG. 10 which are associated with the inductanceL, i.e. the losses of the coil itself plus the equivalent resistance ofthe gate S (FIG. 11). This additional energy loss will thus correspondto the ratio between the capacitances C and C C being of course smallerthan C when positive elements are considered. But the important resultis achieved that this overall transmission loss being flat there will beno distortion depending on the frequency. The actual ratio between C andC can readily be computed since B in (110) is equal to e appearing in(106) provided w t is equal to 1r. The parameter n is given by (102) andin the particular case where the second term of this expression can beneglected with regard to the first, the equation for C /C is no longertranscendental. In such a case if R is sufficiently large with regard toR one may write defining the ratio between the two capacitances and atthe same time the flat loss.

While formula (109) clearly shows that if losses are present in theresonant transfer circuit so that the value of B departs from -1 it isnevertheless possible to secure a flat transmission characteristic infunction of frequency, by a suitable choice of the ratio betweencapacitances C and C the formula also indicates that if another part ofthe transmission circuit already introduces some distortion in functionof frequency and particularly a response which falls as the frequencyincreases, it will be possible to compensate the latter. Indeed,irrespective of Whether B is different from 1 or not, provided RC1 B 8RC2 is negative, (109) indicates that the response will increase as thefrequency increases so that it would be possible to compensate anopposite effect in another part of the 26 I transmission circuit. Sincethe characteristic defined by formula 109) is only present when thereceiving output capacitance C is unloaded, the arrangement cannot beused in a time division multiplex system with the capacitance C being acommon capacitance for the various time slots using the same highway.But the arrangement can nevertheless be used in time division multiplexsystems provided an individual capacitance C is retained for each timeslot. In other words, in addition to the gate S shown in FIG. 11 on theleft of the capacitance C a second gate (not shown) must be provided onthe right of the capacitance C in order that the voltage sample storedacross C can be communicated to a common high input impedance devicesuch as a PCM coder.

Even if another part of the transmission circuit does not introducefrequency distortion, the arrangement of FIG. 11 might also be founduseful if frequency distortion must deliberately be introduced at somepart of the transmission system. Such a case is described in the US.Patent No. 2,621,251 previously referred to at the beginning of thedescription, where pre-emphasis is introduced at the coding end of a PCMtransmission in order to reduce quantization noise. This may be providedby the arrangement of FIG. 11 provided a separate capacitance C isallotted to each time slot. Considering the denominator of the formula(109) it is a linear function of a sin 92:

and may be written as where A is a constant to which a predeterminednegative value may be assigned provided 01 B s RC2 is negative. Thus,recalling (16) and (17), when B is equal to 1, the constant A may bewritten as and the required pre-emphasis can be secured with C C At thereceiving side of the system an arrangement similar to that of FIG. 11may be foreseen but this time, the factor corresponding to A should benegative, e.g. C should be larger than C if there are no losses in theresonant transfer circuit. Again care will be taken to provideindividual capacitances C per time slot. These may for instance beconnected via an individual gate (not shown) to a common high inputimpedance amplifier. In such a manner, with pre-emphasis at the sendingend and de-emphasis at the receiving end, the overall frequencycharacteristic of the system may be defined by T A S111 e) Hm (lA sin TlA sin A sin w T A(l-As1n (115) This same deviation but this time belowunity will be encountered when the angular frequency reaches the valueWhile the above method to secure pre-emphasis will necessitate separatecapacitances C per time slot as well as an individual gate for suchcapacitances if they are to be connected to a common high inputimpedance device, frequency response shaping devices individual perstation are avoided.

It will be clear that although capacitances have been indicated asreactive energy storage devices, inductive elements might eventually beused especially when multiplex circuits are not used and that the rulesof duality, e.g. voltage interchanged with current, series connectioninterchanged With parallel connection, impedance interchanged withadmittance, etc., are then applicable.

While the principles of the invention have been described above inconnection with specific apparatus, it is to be clearly understood thatthis description is made only by way of example and not as a limitationon the scope of the invention.

What is claimed is:

1. A resonant transfer network for transferring energy under the controlof time controlled gates to reactive loads from energy source meansincluding internal resistance means,

first reactance means connected to said energy source for storing acertain amount of energy received from said energy source,

said first reactance means having a first reactive value when viewedfrom the input of said network, second reactance means for receiving,storing and transferring said certain amount of energy,

said second reactance means coupled to a high impedance load wherebysaid second reactance means is unloaded, said second reactance meanshaving a second reactive value when viewed from the output of saidnetwork, said second reactive value being smaller than said firstreactive value, and

interconnecting means including highway means having an input to outputamplitude ratio that is no greater than unity for repeatedlyinterconnecting the first and second reactance means through saidnetwork for effecting the transfer of the certain amount of energystored in said first reactance means to said second reactance means.

2. The resonant transfer network of claim 1 wherein said first andsecond reactance means comprises first and second capacitancesrespectively,

means for permanently interconnecting said capacitances at one terminal,wherein said interconnecting means comprises means interconnecting saidcapacitances at the other terminal for an effective interconnecting timeperiod,

said second capacitance being smaller than said first capacitance, and

wherein the negative of the ratio of said capacitances is equal to theratio of the difference of the voltage across said capacitances at theend of said effective time period and said difference at the start ofsaid time period,

whereby a fiat frequency response in the transfer characteristic betweensaid first and second capacitances is provided.

3. The resonant transfer network of claim 2 wherein the ratio of saidsecond capacitance divided by said first capacitance is equal to ewherein n is the negative real part of complex conjugate roots of theresonant transfer network and t represents the effective interconnectingtime period, and wherein w t =1r.

4. The resonant transfer network of claim 1 wherein the differencebetween said first reactive value and said second reactive valueprovides a frequency dependent response in the transfer characteristicbetween said first and said second reactance means,

said response being inversely proportional to a linear function of wherew is the angular frequency and T is the repetition period of theeffective interconnecting time RALPH D. BLAKESLEE, Primary Examiner

